Question: All of the 4th grade teachers and students from Covington went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$3.00$ each for students, and the group paid $$32.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$22.00$ each for teachers and $$7.50$ each for students, and the group paid $$96.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+3y = 32}$ ${22x+7.5y = 96.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-22x-12y = -128}$ ${22x+7.5y = 96.5}$ Add the top and bottom equations together. $ -4.5y = -31.5 $ $ y = \dfrac{-31.5}{-4.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {5.5x+3y = 32}$ to find $x$ ${5.5x + 3}{(7)}{= 32}$ $5.5x+21 = 32$ $5.5x = 11$ $x = \dfrac{11}{5.5}$ ${x = 2}$ You can also plug ${y = 7}$ into $ {22x+7.5y = 96.5}$ and get the same answer for $x$ ${22x + 7.5}{(7)}{= 96.5}$ ${x = 2}$ There were $2$ teachers and $7$ students on the field trips.